On a $q$-deformed Fock space, we define multiple $q$-LÚvy processes. Using the partition-dependent stochastic measures derived from such processes, we define partition-dependent cumulants for their joint distributions, and express these in terms of the cumulant functional using the number of restricted crossings of P. Biane. In the single variable case, this allows us to define a $q$-convolution for a large class of probability measures. We make some comments on the It˘ table in this context, and investigate the $q$-Brownian motion and the $q$-Poisson process in more detail.